| 252 |
SI unit for linear momentum
|
\[
\mathrm{kg}\cdot\frac{\mathrm{m}}{\mathrm{s}}
\]
|
| 252 |
linear momentum
of a particle of mass $m$ with velocity $\vect{v}$
|
\[
\vect{p}\equiv m\vect{v}
\]
|
| 252 |
linear momentum components
in $x, y, \text{ and } z$ directions
|
\[
\begin{align*}
p_x&=mv_x\\
p_y&=mv_y\\
p_z&=mv_z
\end{align*}
\]
|
| 253 |
net force
on a particle with linear momentum $\vect{p}$
|
\[
\sum\vect{F}=\frac{d\vect{p}}{dt}
\]
|
| 254 |
law of conservation of linear momentum, two-particle system
|
\[
\begin{align*}
\vect{p}_{1i}+\vect{p}_{2i}
&=\vect{p}_{1f}+\vect{p}_{2f}\\
\vect{p}_{\mathrm{tot}}
&=\sum_{\mathrm{system}}\vect{p}\\
&=\vect{p}_1+\vect{p}_2\\
&=\mathrm{constant}
\end{align*}
\]
|
|
Momentum is independently conserved
in
the $x,\ y, \text{ and } z$ directions.
|
\[
\begin{align*}
\sum_{\mathrm{system}} p_{i_x}
&=\sum_{\mathrm{system}} p_{f_x}\\
\sum_{\mathrm{system}} p_{i_y}
&=\sum_{\mathrm{system}} p_{f_y}\\
\sum_{\mathrm{system}} p_{i_z}
&=\sum_{\mathrm{system}} p_{f_z}
\end{align*}
\]
|
|
impulse.
impulse momentum theorem.
The impulse of the force $\vect{F}$
acting on a particle equals the change
in the momentum of the particle caused
by that force.
|
\[
\vect{I}\equiv\int_{t_i}^{t_f}\vect{F}\,dt\equiv\Delta\vect{p}
\]
|
|
time-averaged force
|
\[
\overline{\vect{F}}\equiv\frac{1}{\Delta t}\int_{t_i}^{t_f}\vect{F}\,dt
\]
|
|
impulse in terms of time-averaged force
|
\[
\vect{I}\equiv\overline{\vect{F}}\Delta t
\]
|
|
impulse when impulsive force is constant
|
\[
\vect{I}\equiv\overline{\vect{F}}\Delta t,\ \ \vect{F}=\mathrm{constant}
\]
|
|
perfectly inelastic collision.
total momentum before the collision
equals the total momentum of the
composite system after the collision.
|
\[
m_1\vect{v}_{1i}+m_2\vect{v}_{2i}=\left(m_1+m_2\right)\vect{v}_f
\]
|
|
center of mass of a system of particles
where the $i\text{th}$ particle has mass
$m_i$ and position $\vect{r}_i.$
|
\[
\vect{r}_{\mathrm{CM}}\equiv\frac{1}{M}\sum_{i}{m_i\vect{r}_i}
\]
|
|
center of mass of a rigid body
|
\[
\vect{r}_{\mathrm{CM}}=\frac{1}{M}\int\vect{r}\,dm
\]
|
|
velocity of the center of mass for a system of particles
|
\[
\begin{align*}
\vect{v}_{\mathrm{CM}}
&=\frac{1}{M}\sum_{i}{m_i\vect{v}_i}\\
&=\frac{d\vect{r}_{\mathrm{CM}}}{dt}\\
&=\frac{1}{M}\sum_{i}{m_i\frac{d\vect{r}_i}{dt}}
\end{align*}
\]
|
|
total linear momentum of a system of particles
|
\[
\vect{p}_{\mathrm{tot}}=M\vect{v}_{\mathrm{CM}}=\sum_{i}{m_i\vect{v}_i}=\sum_{i}\vect{p}_i
\]
|
|
acceleration of the center of mass of a system of particles
|
\[
\begin{align*}
\vect{a}_{\mathrm{CM}}
&=\frac{1}{M}\sum_{i}{m_i\vect{a}_i}\\
&=\frac{d\vect{v}_{\mathrm{CM}}}{dt}\\
&=\frac{1}{M}\sum_{i}{m_i\frac{d\vect{v}_i}{dt}}
\end{align*}
\]
|
|
resultant external force on a system of particles
|
\[
\sum\vect{F}_{\mathrm{ext}}=M\vect{a}_{\mathrm{CM}}=\frac{d\vect{p}_{\mathrm{tot}}}{dt}
\]
|
|
law of conservation of momentum for a system of particles
|
\[
\sum\vect{F}_{\mathrm{ext}}=0\Rightarrow\vect{p}_{\mathrm{tot}}=\mathrm{constant}
\]
|
Alex Notes